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Nov 18, 1974 - Olagoke Olabisi and Robert Simha*. Department of Macromolecular Science, Case Western Reserve University,. Cleveland, Ohio 44106.
Theoretical Considerations of Amorphous Polymers

Vol. 8, No. 2, March-April 1975

21 1

Configurational Thermodynamic Properties of Amorphous Polymers and Polymer Melts. 11. Theoretical Considerations Olagoke Olabisi and Robert Simha* Department of Macromolecular Science, Case Western Reserve University, Cleueland, Ohio 44106. Received November 18,1974

ABSTRACT: The three methacrylate polymers and melts of low and high density polyethylenes investigated in the preceding paper are discussed in terms of theory. Corresponding literature data on n-paraffins and hevea rubbers are also considered. The good agreement between experimental and predicted PVT relations obtained for the high polymers is similar to that found earlier in several instances. The extensive results available a t present make possible comparisons of the characteristic scaling parameters for different systems, with a variation of the characteristic temperatures by a factor of 2. A relationship between the characteristic compressibility factor or entropy per unit mass and the temperature scaling factor ensues, which results in a correlation between the characteristic segmental energy factor and the two-thirds power of the segmental‘mass. Proceeding to the pressure dependence of the thermodynamic functions, we find again good agreement between experimental and theoretical energies andentropies. The results once more illustrate the inadequacy of a van der Waals form for the configurational internal energy. The analysis of the liquid-glass transition line in the methacrylate systems yields the variation of the hole fraction 1 - y along the boundary for the glasses formed by a variable pressure history, and a satisfactory constancy for the glass formed under atmospheric pressure. From a combination of the theoretical and the experimental equation of state of the latter we derive the temperature and pressure dependence of y and compute an internal energy without as defined earlier, increases significantly with pressure in further adjustments. Whereas the frozen fraction a t Tg, polystyrene, it remains nearly constant in the methacrylates. From the magnitude of the hole fractions (or free volumes) it follows that a significant extent of hole clustering should occur in high Tgsystems.

In a series of recent papers we have compared experimental results originating from our laboratory or the literature with theoretical predictions. These have included polymers of styrene (PS) and o-methylstyrene (PoMS),’ atactic and isotactic methyl methacrylate (PMMA),2J vinyl chloride (PVC),* and vinyl acetate (PVAC).4 The results and a detailed examination of dilatometric data at atmospheric pressure516 revealed the good agreement between the experimental and theoretical5 equations of state. This enabled us to examine the liquid-glass transition region in terms of the equilibrium t h e ~ r y ,and ~ , ~to predict the pressure dependence of the glass temperature. Finally we have explored the equation of state of the glass itself and the modifications required in the equilibrium theory and the temperature and pressure dependence of the ordering parameter appearing in the theory.4.7 We wish to pursue these directions with the polymers investigated in the preceding paper.s The theoretical description of a particular liquid system proceeds in terms of characteristic volume ( V * ) ,temperature ( T * ) , and pressure (P*) scaling parameter^,^ which are to reflect structural characteristics of the segment within, of course, the assumptions of the theory. The analysis of the recent liquid state results8 will enlarge the collection of such parameter values, besides providing the intrinsic information. Considering possible correlations between these quantities,g the methacrylates will furnish further examples of relatively high and intermediate T , systems. The polyethylene melts, on the other hand, represent examples of low T,’s. The scale of T*’s will be further extended by the inclusion of n-alkanes. 1 0 ~ Finally we examine the liquid-glass transition zone, the glassy state of the methacrylate polymers, and the comparative behavior of the ordering parameter in the glass.

I. Equation of State The relations to be employed are in reduced variables5 (23’/?) (y?jm2[ 1.01 1 (),?)-’- 1.20451 (1) The hole fraction 1 - y satisfies the equilibrium condition for an infinite s-mer

v,

The (numerical) solution of eq 2 yields y as a function of T, and the “flexibility” ratio 3c/s, ie., the number of volume-dependent degrees of freedom per chain segment. For a long chain with a large number of modes of motion this ratio, to be meaningful, must be of the order of unity. The actual value should be a characteristic of the particular system. We have no independent knowledge of this ratio and it cannot be obtained from PVT data without additional assumptions or information, since the appropriate reduced functions, computed for different numerical values, are superimpo~able.~ In previous applications we have assigned a universal value of unity to the flexibility parameter. The temperature T* is defined5 for a quasilattice of coordination number z by the ratio ( z - P)st*/(ch) and then becomes directly proportional to the intersegmental potential minimum e * . Moreover, the relation between the effective segment in eq 1 and 2 and the chemical repeat unit will be affected by the requirement 3c/s = 1. That is, a simple structure such as a polymethylene should require a comparatively larger segmental unit than for example polystyrene. From the definitions of the scaling parameters (recall that P* is proportional to the ratio se*/(su*), with u* the segmental hard core volume), combined with the above assumption, we have the relation2 (P*I*/T*).Uo = R ( c / s ) = 27.7

(3)

where V* is expressed in cm3/g and Mo is the molecular weight of the segment. Since sMo = (dp)M(monomer), d p is the degree of polymerization, there follows c/dp = (P Lm/T*)M(monomer)/R

(3’)

11. Evaluation of Scaling Parameters A. Atmospheric Pressure. Figure 1 summarizes the volume-temperature results for our three polyethylene melts8 and Hellwege, et al.’s, samples.12 From the superposition of experimental data on the theoretical line, V* and T* are

212 Olabisi, Simha

Macromolecules

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Figure 1. Reduced volume as a function of reduced temperatufe for polyethylenes at atmospheric pressure.

0.90

1.00

0.95

p

Figure 3. Reduced compressibility factor as a function of reduced density at a series of temperatures for PCHMA. I

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tIneor Polyethylene

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103

% Figure 2. Reduced compressibility factor as a function of reduced density at a series of temperatures for PnBMA. 0.86

obtained. The agreement noted in Figure 1 is typical of that found earlier for other polymers and the methacrylates studied in this series,s and for hevea rubber and its vulcanizates, as analyzed by Olabisi.13 A more severe test of the theory involves an examination of the thermal expansivities.6 The theoretical isobar in Figure 1 can be accurately represented by a combination of eq 1 when P = 0, with eq 2 by means of the interpolation formula2 In (?/0.90181)

or

= 23.834,(?)3'2

=

('h)zP

w

CY = 35.751,(?)1'2 (4) in the range 1.65 < T X 102 < 7.03. Slightly different numerical coefficients over a narrower temperature interval have been derived recently.4 The differences between experimental and theoretical expansivities are less than 10%.6 However, eq 4 predicts in general too strong a temperatur? dependence of cy. These departures of the theoretical &-T relation can be translated into departwes of V* and T* from the average value derived by the superposition procedure. Insertion of the experimental a, V , and T values into eq 4 provides a point by point determination of V* and T*.

0.94

0.90

0.98

Yo Figure 4. Reduced compressibility factor as a function of reduced density at a series of temperatures for LPE ( p = 0.9794 g/cm3). In this manner we observe a maximum increase in T* and V* of 5 and 0.3% respectively with increasing T,13based on a range of looo, comparable again with earlier findings.6 B. Elevated Pressure. With V* and T* determined, P* is obtained by comp_arLng the theoretical reduced compressibility factor PVIT with the semireduced quantity PP/.-i'.The resulting P* values are averaged over all experimental isotherms. The outcome of this procedure for PnBMA and PCHMA is illustrated in Figures 2 and 3. Due to the low T,,the widest range of temperatures is available for PnBMA. Deviations from the good agreement between experiment and theory, noticeable otherwise, appear at thehighest temperature (199.5'). A similar pattern obtains for PCHMA. The increasing shortening of the pressure range with decreasing temperature is, of course, due to the intervention of the glass transition. The isotherms for the three polyethylenes are shown in Figures 4-6. The results for Hellwege, e t al.'s, polymers1* are very similar.l3 The comparatively small values of T* result in the highest reduced temperatures for polymers

Theoretical Considerations of Am'orphous Polymers 213

Vol. 8, No. 2, March-April 1975 I

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6

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12

4

8

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7

e

Ehptl.

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0 0.88

0.92

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0.96

1.00

Figure 5. Reduced compressibility factor as a function of reduced density at a series of temperatures for BPE ( p = 0.9320 g/cm3).

Figure 7. Reduced compressibility factor as a function of reduced density at a series of temperatures for n-tridecane.

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0 0.84

0.88

0.92

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Figure 6. Reduced compressibility factor as a function of reduced density a t a series of temperatures for HMLPE ( p = 0.9268 g/cm3).

Figure 8. Reduced compressibility factor as a function of reduced density at a series of temperatures for n-tetracontane.

studied so far. In the upper temperature and pressure ranges the theoretical compressibility factor tends to be smaller than the experimental one. Since we are dealing with essentially a low-temperature theory, even with the introduction of holes, this trend is consistent. Figures 7 and 8 describe Doolittle's data on n-alkanes.lOJ1 Equations 1 and 2 have been employed without corrections for end effects. This is not sensibly reflected in the results at atmospheric pressure. However, at elevated pressures the agreement between experiment and theory is decidely poorer for the two alkanes, although again better at the lower temperatures. No definite chain-length effect can be extracted from Figures 7 and 8. Finally, some literature